A vector relates two given points. It is a mathematical quantity having both the Magnitude and the direction.

## Multiplication of Vectors

Multiplication of vectors can be of two types:

**(i) Scalar Multiplication**

**(ii) Vector Multiplication**

Here, we will discuss only about the Scalar Multiplication by

__Multiplication of vectors with scalar:__

When a vector is multiplied by a scalar quantity, then the magnitude of the vector changes in accordance with the magnitude of the scalar but the direction of the vector remains unchanged.

Suppose we have a vector \( \overrightarrow {a} \)*k* then we get a new vector with magnitude as |\( \overrightarrow {ka} \)*k* is positive and if *k* is negative then the direction of *k* becomes just opposite of the direction of vector \( \overrightarrow {a} \)**. **

Now let us understand visually the scalar multiplication of vector

Let us take the values of ‘k ‘to be = 2,3,-3,\( \frac {-1}{2}\)

From the above given set of vectors we see that the direction of vector \( \overrightarrow {a} \)

As per above discussions we can see that

\(~~~~~~~~\)

Suppose if the value of the scalar multiple *k *is -1 then by scalar multiplication we know that resultant vector is \( \overrightarrow {-a} \)**, **then \( \overrightarrow {a} \)**. **The vector \( \overrightarrow {-a} \)**. **

Now suppose the value of *k = *\( \frac {1}{|a|} \)

Also, as per above discussion, if *k = *0 then the vector also becomes zero.

Let us go through an example to make this point more clear,

**Example:** **A vector is represented in orthogonal system as \( \overrightarrow {a} \) = \( 3 \hat i + \hat j + \hat k \) . What would be the resultant vector if \( \overrightarrow {a} \) is multiplied by 5 ?**

**Solution:** As the vector is to be multiplied by a scalar the resultant would be,

5 \( \overrightarrow {a} \)

\( \overrightarrow {5a} \)

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